COMEDK 2023 Morning Shift

The slope of the tangent to the curve, \(y=x^2-x y\) at \(\left(1, \frac{1}{2}\right)\) is

Let \(f(x)=a+(x-4)^{\frac{4}{9}}\), then minima of \(f(x)\) is

Using mathematical induction, the numbers \(a_n \delta\) are defined by \(a_0=1, a_{n+1}=3 n^2+n+a_n, (n \geq 0)\). Then, \(a_n\) is equal to

If \(49^n+16^n+k\) is divisible by 64 for \(n \in N\), then the least negative integral value of \(k\) is

\(2^{3 n}-7 n-1 \text { is divisible by }\)

The total number of terms in the expansion of \((x+y)^{60}+(x-y)^{60}\) is

The coefficient of \(x^{29}\) in the expansion of \(\left(1-3 x+3 x^2-x^3\right)^{15}\) is

In the expansion of \(\left(1+3 x+3 x^2+x^3\right)^{2 n}\), the term which has greatest binomial coefficient, is

The points of intersection of circles \((x+1)^2+y^2=4\) and \((x-1)^2+y^2=9\) are \((a, \pm b)\), then \((a, b)\) equals to

The circle \(x^2+y^2+3 x-y+2=0\) cuts an intercept on \(X\)-axis of length

\(S \equiv x^2+y^2-2 x-4 y-4=0\) and \(S^{\prime} \equiv x^2+y^2-4 x-2 y-16=0\) are two circles the point \((-2,-1)\) lies

If \(z=\sqrt{3}+i\), then the argument of \(z^2 e^{z-i}\) is equal to

If \(i=\sqrt{-1}\) and \(n\) is a positive integer, then \(i^n+i^{n+1}+i^{n+2}+i^{n+3}\) is equal to

If \(\left(\frac{3}{2}+i \frac{\sqrt{3}}{2}\right)^{50}=3^{25}(x+i y)\), where \(x\) and \(y\) are real, then the ordered pair \((2 x, 2 y)\) is

\(\int\limits_{-\pi / 2}^{\pi / 2} \sin ^2 x d x\) is equal to

The differential equation of all non-vertical lines in a plane is

The general solution of \(\left(\frac{d y}{d x}\right)^2=1-x^2-y^2+x^2 y^2\) is

The solution of the differential equation \(\left(\frac{d y}{d x}\right) \tan y=\sin (x+y)+\sin (x-y)\) is

The approximate value of \(f(5.001)\), where \(f(x)=x^3-7 x^2+10\)

If \(2 f\left(x^2\right)+3 f\left(\frac{1}{x^2}\right)=x^2-1, \forall x \in R-\{0\}\), then \(f\left(x^8\right)\) is equal to

\(\int \frac{x d x}{2(1+x)^{3 / 2}}\) is equal to

\(\int \frac{4^x}{\sqrt{1-16^x}} d x\) is equal to

The value of \(\lim _\limits{x \rightarrow 0} \frac{e^{a x}-e^{b x}}{2 x}\) is equal to

If \(f(x) = \left\{ {\matrix{ {2\sin x} & ; & { - \pi \le x \le {{ - \pi } \over 2}} \cr {a\sin x + b} & ; & { - {\pi \over 2} < x < {\pi \over 2}} \cr {\cos x} & ; & {{\pi \over 2} \le x \le \pi } \cr } } \right.\) and...

The value of \(\lim _\limits{x \rightarrow \infty}\left(\frac{x^2-2 x+1}{x^2-4 x+2}\right)^{2 x}\) is

The feasible region for the inequations \(x+2 y \geq 4,2 x+y \leq 6, x, y \geq 0\) is

The maximum value of \(Z=10 x+16 y\), subject to constraints \(x \geq 0, y \geq 0, x+y \leq 12,2 x+y \leq 20\) is

The maximum value of \(Z=12 x+13 y\), subject to constraints \(x \geq 0, y \geq 0, x+y \leq 5\) and \(3 x+y \leq 9\) is

The value of \(a^{\log _b c}-c^{\log _b a}\), where \(a, b, c>0\) but \(a, b, c \neq 1\), is

If $$A=\left[\begin{array}{ll}2 & 2 \\ 3 & 4\end{array}\right]$$, then \(A^{-1}\) equals to

If \(A\) is a matrix of order \(4 \operatorname{such}\) that \(A(\operatorname{adj} A)=10 \mathrm{~I}\), then \(|\operatorname{adj} A|\) is equal to

If $$A=\left[\begin{array}{cc}k+1 & 2 \\ 4 & k-1\end{array}\right]$$ is a singular matrix, then possible values of \(\mathrm{k}\) are

\(\text { Find }{ }^n C_{21} \text {, if }{ }^n C_{10}={ }^n C_{12}\)

There are 10 points in a plane out of which 4 points are collinear. How many straight lines can be drawn by joining any two of them?

The total number of numbers greater than 1000 but less than 4000 that can be formed using 0, 2, 3, 4 (using repetition allowed) are

A polygon of n sides has 105 diagonals, then n is equal to

A number \(\mathrm{n}\) is chosen at random from \(s=\{1,2,3, \ldots, 50\}\). Let \(\mathrm{A}=\{n \in s: n\) is a square \(\}\), \(\mathrm{B}=\{n \in s: n\) is a prime\(\}\) and \(\mathrm{C}=\{n \in s: n\) is a square\(\}\). Then, correct ...

A five-digits number is formed by using the digits \(1,2,3,4,5\) with no repetition. The probability that the numbers 1 and 5 are always together, is

If a number $n$ is chosen at random from the set \(\{11,12,13, \ldots \ldots, 30\}\). Then, the probability that \(n\) is neither divisible by 3 nor divisible by 5, is

Three vertices are chosen randomly from the nine vertices of a regular 9-sided polygon. The probability that they form the vertices of an isosceles triangle, is

If \(A, B\) and \(C\) are mutually exclusive and exhaustive events of a random experiment such that \(P(B)=\frac{3}{2} P(A)\) and \(P(C)=\frac{1}{2} P(B)\), then \(P(A \cup C)\) equals to

In a trial, the probability of success is twice the probability of failure. In six trials, the probability of at most two failure will be

The sum of \(n\) terms of the series, \(\frac{4}{3}+\frac{10}{9}+\frac{28}{27}+\ldots\) is

The value of \(\frac{1}{2 !}+\frac{2}{3 !}+\ldots+\frac{99}{100 !}\) is equal to

If the sum of 12th and 22nd terms of an AP is 100, then the sum of the first 33 terms of an \(\mathrm{AP}\) is

If \(A=\{a, b, c\}, B=\{b, c, d\}\) and \(C=\{a, d, c\}\) then \((A-B) \times(B \cap C)\) is equal to

If \(n(A)=p\) and \(n(B)=q\), then the numbers of relations from the set \(A\) to the set \(B\) is

Let the equation of pair of lines \(y=m_1 x\) and \(y=m_2 x\) can be written as \(\left(y-m_1 x\right)\left(y-m_2 x\right)=0\). Then, the equation of the pair of the angle bisector of the line \(3 y^2-5 x y-2 x^2=0\) is

The distance of the point \((3,4)\) from the line \(3 x+2 y+7=0\) measured along the line parallel to \(y-2 x+7=0\) is equal to

The slope of lines which makes an angle \(60^{\circ}\) with the line \(y-3 x+18=0\)